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Meaning of Poisson's ratio

[Rod Lakes, University of Wisconsin]

Definition of Poisson's ratio
Poisson's ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force. Tensile deformation is considered positive and compressive deformation is considered negative. The definition of Poisson's ratio contains a minus sign so that normal materials have a positive ratio. Poisson's ratio, also called Poisson ratio or the Poisson coefficient, is usually represented as a lower case Greek nu, n.
If your browser does not interpret Symbol font properly, Greek nu, n may instead look like a bold face Latin n. For an alternative page go here.

stretch animation
n = - etrans / elongitudinal

Strain e is defined in elementary form as the change in length divided by the original length.
e = DL/L.

Poisson's ratio: why usually positive
Virtually all common materials, such as the blue rubber band on the right, become narrower in cross section when they are stretched. The reason why, in the continuum view, is that most materials resist a change in volume as determined by the bulk modulus K more than they resist a change in shape, as determined by the shear modulus G.
stretch honeycomb In the structural view, the reason for the usual positive Poisson's ratio is that inter-atomic bonds realign with deformation. Stretching of yellow honeycomb by vertical forces, shown on the right, illustrates the concept.

Poisson's ratio: relation to elastic moduli
Poisson's ratio is related to elastic moduli K, the bulk modulus; G as the shear modulus; and E, Young's modulus, by the following. The elastic moduli are measures of stiffness. They are ratios of stress to strain. Stress is force per unit area, with the direction of both the force and the area specified.
n = (3K - 2G)/(6K + 2G)
E = 2G( 1 + n)

The theory of isotropic elasticity allows Poisson's ratios in the range from -1 to 1/2 for an object with free surfaces with no constraint. Physically the reason is that for the material to be stable, the stiffnesses must be positive; the bulk and shear stiffnesses are interrelated by formulae which incorporate Poisson's ratio. Objects constrained at the surface can have a Poisson's ratio outside the above range and be stable.

Poisson's ratio in various materials
Poisson's ratio of the elements are via Web Elements, which adduce references [2-4]. In a large compilation of properties of polycrystalline materials [5], most have Poisson's ratio in the vicinity of 1/3.
Material
Isotropic upper limit [1]
Rubber [6]
Lead
Copper [7]
Aluminum
Copper
Polystyrene [6]
Brass [1]
Ice [8]
Polystyrene foam [6]
Stainless Steel [7]
Steel [1]
Beryllium
Re-entrant foam [9]
Isotropic lower limit [1] 		Poisson's ratio
0.5
0.48- ~0.5
0.44
0.37
0.35
0.34
0.34
0.33
0.33
0.3
0.30
0.29
0.08
-0.7
-1 		References
[1] I. S. Sokolnikoff, Mathematical theory of elasticity. Krieger, Malabar FL, second edition, 1983.
[2] A .M. James and M. P. Lord in Macmillan's Chemical and Physical Data, Macmillan, London, UK, 1992.
[3] G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993.
[4] G.V. Samsonov (Ed.) in Handbook of the physicochemical properties of the elements, IFI-Plenum, New York, USA, 1968.
[5] G. Simmons, and H. Wang, Single crystal elastic constants and calculated aggregate properties: a handbook, MIT Press, Cambridge, 2nd ed, 1971.
[6] J. A. Rinde, Poisson's ratio for rigid plastic foams, J. Applied Polymer Science, 14, 1913-1926, 1970.
[7] D. E. Gray, American Institute of Physics Handbook, 3rd ed., chapter 3, McGraw hill, New York, 1973.
[8] E. M. Schulson, The Structure and Mechanical Behavior of Ice, JOM, 51 (2) pp. 21-27, 1999. article link
[9] R. S. Lakes, Foam structures with a Negative Poisson's ratio, Science, 235 1038-1040, 1987.


Poisson's ratio in bending.
Bend a bar or plate. Poisson's ratio governs the curvature in a direction perpendicular to the direction of bending. This "anticlastic curvature" is easily seen in the bending of a rubber eraser. Shown here is bending, by a moment applied to opposite edges, of a honeycomb with hexagonal cells. If the honeycomb cells are regular hexagons, the Poisson's ratio can approach +1. Since the honeycomb is anisotropic, the Poisson's ratio need not lie within the above range. bend honeycomb

Poisson's ratio in viscoelastic materials
The Poisson's ratio in a viscoelastic material is time dependent in the context of transient tests such as creep and stress relaxation. If the deformation is sinusoidal in time, Poisson's ratio may depend on frequency, and may have an associated phase angle. Specifically, the transverse strain may be out of phase with the longitudinal strain in a viscoelastic solid. Get pdf of a research article on this.

Poisson's ratio, waves and deformation
The Poisson's ratio of a material influences the speed of propagation and reflection of stress waves. In geological applications, the ratio of compressional to shear wave speed is important in inferring the nature of the rock deep in the Earth. This wave speed ratio depends on Poisson's ratio. Poisson's ratio also affects the decay of stress with distance according to Saint Venant's principle, and the distribution of stress around holes and cracks.

An example. Analysis of effect of Poisson's ratio on compression of a layer.
What about the effect of Poisson's ratio on constrained compression in the 1 (or x) direction? Constrained compression means that the Poisson effect is restrained from occurring. This could be done by side walls in an experiment. Also, compression of a thin layer by stiff surfaces is effectively constrained. Moreover, in ultrasonic testing, the wavelength of the ultrasound is usually much less than the specimen dimensions. The Poisson effect is restrained from occurring in this case as well.
In Hooke's law (with the elastic modulus tensor as Cijkl we sum over k and l, but, due to the constraint, the only strain component which is non-zero is e11.
sij = Cijkl ekl = C1111e11 + C1122e22 + C1133e33 = C1111e11 ,

so the effective stiffness for constrained compression is C1111.

Let us find the physical significance of that tensor element in terms of engineering constants.

One may also work with the elementary isotropic form for Hooke's law.
exx = (1/E) { sxx - nsyy - nszz}
eyy =(1/E) { syy - nsxx - nszz}
ezz = (1/E) { szz - nsxx - nsyy}

For simple tension or compression in the x direction, the Poisson effect is free to occur. There is stress in only one direction but there can be strain in three directions. sxx ≠ 0, that is not equal to zero, syy = 0, szz = 0. Then
(sxx / exx) = E.
So Young's modulus E is the stiffness for simple tension, with the Poisson effect free to occur.

Consider constrained compression, with eyy = 0, ezz = 0. Then
syy = nsxx + nszz.
szz = nsxx + nsyy.
Substituting,
syy = szz = sxx ( n(1 + n)/(1 - n2)) .
So, substituting into Hooke's law, the stress-strain ratio for constrained compression, which by definition is the constrained modulus C1111, is
(sxx/exx) = C1111 = E ((1 - n) / (1 + n) (1 - 2n)).

The physical meaning of C1111 is the stiffness for tension or compression in the x (or 1) direction, when strain in the y and z directions is constrained to be zero. The reason is that for such a constraint the sum in the tensorial equation for Hooke's law collapses into a single term containing only C1111. The constraint could be applied by a rigid mold, or if the material is compressed in a thin layer between rigid platens. C1111 also governs the propagation of longitudinal waves in an extended medium, since the waves undergo a similar constraint on transverse displacement.

Rubbery materials have Poisson's ratios very close to 1/2, shear moduli on the order of a MPa, and bulk moduli on the order of a GPa. Therefore the constrained modulus C1111 is comparable to the bulk modulus and is much larger than the shear or Young's modulus of rubber.

Practical example - cork in a bottle.
An example of the practical application of a particular value of Poisson's ratio is the cork of a wine bottle. The cork must be easily inserted and removed, yet it also must withstand the pressure from within the bottle. Rubber, with a Poisson's ratio of 0.5, could not be used for this purpose because it would expand when compressed into the neck of the bottle and would jam. Cork, by contrast, with a Poisson's ratio of nearly zero, is ideal in this application.

Practical example - design of rubber buffers.
How does three-dimensional deformation influence the use of viscoelastic rubber in such applications as shoe insoles to reduce impact force in running, or wrestling mats to reduce impact force in falls?

Solution
Refer to the above analysis, in which deformation under transverse constraint is analyzed. Rubbery materials are much stiffer when compressed in a thin layer geometry than they are in shear or in simple tension; they are too stiff to perform the function of reducing impact. Compliant layers can be formed by corrugating the rubber to provide room for lateral expansion or by using an elastomeric foam, which typically has a Poisson's ratio near 0.3, in contrast to rubber for which Poisson's ratio can exceed 0.49. Corrugated rubber is used in shoe (sneaker) insoles and in vibration isolators for machinery. Foam is used in shoes and in wrestling mats.

squeeze block Practical example - aircraft sandwich panels.
The honeycomb shown above is used in composite sandwich panels for aircraft. The honeycomb is a core between face-sheets of graphite-epoxy composite. Such panels are usually flat. If curved panels are desired, the honeycomb cell shape must be changed from the usual regular hexagon shape, otherwise the cells will be crushed during bending. Several alternative cell shapes are known, including those which result in a negative Poisson's ratio.

Practical example - Interpreting compression tests on blocks of flexible material.
The material is constrained at contact surfaces by the compression device, so the Poisson effect cannot freely occur. Bulge occurs in the middle as shown in the image on the right. Determination of elastic moduli involves use of correction formulae.

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